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Could anyone tell me what is the minimal voltage between anode and cathode in a linear accelerators to achieve speeds where relativity starts to show?

Let me ask in a different way: "What is the minimal voltage scientists work with in linear accelerators?"

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What is a "speed where relativity starts to show?" –  David Z Dec 11 '12 at 22:21

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Well, let's just play an order of magnitude game. Relativity should start to show up at kinetic energies approaching a particle's rest mass. So, we have:

$$\begin{align} qV &= mc^{2}\\ V &= \frac{mc^{2}}{q} \end{align}$$

Using the values for the electron, we get about $5\times 10^{5}$ V for electrons to turn relativistic. Protons are going to take longer, an anything heavier will take yet longer, since $m$ typically grows a lot faster than $q$. If we want to get more precise than this, we are going to need to start talking precisions and things.

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Could anyone tell me what is the minimal voltage between anode and cathode in a linear acellerators to achieve speeds where relativity starts to show?

Define the limits of "starts to show" that you care about.

Electrons are significantly relativistic at 100 kV and measurably so at a few 10s of kV.

Let me ask in a different way: "What is the minimal voltage scientists work with in linear acelerators?"

Every CRT is linear accelerator, so the values go arbitrarily low, and nuclear physics is sometime done at a few tens of keV. It really depends on what you want to probe.

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